η a ¼ 1 À
1
2
η a
1
2
Φ
2
This equation simply gives us a modified absorption rate instead of (3.6.14) in the
form:
η a ¼ 1 þ
1
4
Φ
2
À1 1
2
Φ
2
ð3:8:3Þ
This absorption rate is plotted as a function of τ in Fig. 3.29 with dotted line. It is
noted that this simple modification gives almost the same result. Suck kind of
feedback to an external parameter in a simplified model is called pump depletion
in general.
3.9 Large Amplitude Electron Plasma Waves
Eliminating the driver term in Eq. (3.6.4), the equation tends to an exact equation for
nonlinear plasma wave at T ¼ 0:
d
2
dt
2
V þ ω
2
pe0 V ¼ 0
ð3:9:1Þ
Introducing the Lagrange coordinate x 0 defined by:
x 0 ¼ x À
Z t
0
V x 0 , τ
ð
Þdτ
ð3:9:2Þ
It is easy to obtain the nonlinear plasma wave structure. The plasma wave propagating to the positive direction in a uniform density has a solution of (3.9.1) in the
form:
V ¼ V 0 sin kx 0 À ω p0 t
À
Á
ð3:9:3Þ
Inserting Eq. (3.9.3) to Eq. (3.6.3) and taking integration gives:
E ¼
V 0
ω p0
cos kx 0 À ω p0 t
À
Á
ð3:9:4Þ
The density profile can be solved with use of the continuity relation:
118
3 Ultra-Short Pulse and Collisionless Absorption